Negative local resistance due to viscous electron backflow in graphene

Negative local resistance due to viscous electron backflow in graphene

D. A. Bandurin ¹ , I. Torre ^2,3 , R. Krishna Kumar ^1,4 , M. Ben Shalom ^1,5 , A. Tomadin ⁶ , A. Principi ⁷ , G. H. Auton ⁵ , E. Khestanova ^1,5 , K. S. NovoseIov ⁵ , I. V. Grigorieva ¹ , L. A. Ponomarenko ^1,4 , A. K. Geim ¹ , M. Polini ³

1 School of Physics & Astronomy, University of Manchester, Oxford Road, Manchester M13 9PL, United Kingdom

2 National Enterprise for nanoScience and nanoTechnology, Scuola Normale Superiore, I-56126 Pisa, Italy

3 Istituto Italiano di Tecnologia, Graphene labs, Via Morego 30 I-16163 Genova (Italy)

4 Physics Department, Lancaster University, Lancaster LA14YB, United Kingdom

5 National Graphene Institute, University of Manchester, Manchester M13 9PL, United Kingdom

6 National Enterprise for nanoScience and nanoTechnology, Istituto Nanoscienze-Consiglio Nazionale delle Ricerche and Scuola Normale Superiore, I-56126 Pisa, Italy

7 Radboud University, Institute for Molecules and Materials, NL-6525 AJ Nijmegen, The Netherlands

Graphene hosts a unique electron system in which electron-phonon scattering is extremely weak but electron-electron collisions are sufficiently frequent to provide local equilibrium above liquid nitrogen temperature. Under these conditions, electrons can behave as a viscous liquid and exhibit hydrodynamic phenomena similar to classical liquids. Here we report strong evidence for this long-sought transport regime. In particular, doped graphene exhibits an anomalous (negative) voltage drop near current injection contacts, which is attributed to the formation of submicrometer-size whirlpools in the electron flow. The viscosity of graphene’s electron liquid is found to be ≈0.1 m ² s ^-1 , an order of magnitude larger than that of honey, in agreement with many-body theory. Our work shows a possibility to study electron hydrodynamics using high quality graphene.

Collective behavior of many-particle systems that undergo frequent inter-particle collisions has been

studied for more than two centuries and is routinely described by the theory of hydrodynamics (1,2). The

theory relies only on the conservation of mass, momentum and energy and is highly successful in

explaining the response of classical gases and liquids to external perturbations varying slowly in space

and time. More recently, it has been shown that hydrodynamics can also be applied to strongly

interacting quantum systems including ultra-hot nuclear matter and ultra-cold atomic Fermi gases in the

unitarity limit (3-6). In principle, the hydrodynamic approach can also be employed to describe

many-electron phenomena in condensed matter physics (7-13). The theory becomes applicable if

electron-electron scattering provides the shortest spatial scale in the problem such that ℓ ee ≪ 𝑊𝑊, ℓ

where ℓ ee is the electron-electron scattering length, 𝑊𝑊 the characteristic sample size, ℓ ≡ 𝑣𝑣 F 𝜏𝜏 the

mean free path, 𝑣𝑣 the Fermi velocity, and 𝜏𝜏 the mean free time with respect to

inequalities are difficult to meet experimentally. Indeed, at low temperatures ( 𝑇𝑇 ) ℓ ee varies approximately as ∝ 𝑇𝑇 ⁻² reaching a micrometer scale at liquid-helium 𝑇𝑇 (14), which necessitates the use of ultra-clean systems to satisfy ℓ ee ≪ ℓ. At higher 𝑇𝑇, electron-phonon scattering rapidly reduces ℓ. However, for two-dimensional (2D) systems with dominating acoustic phonon scattering, ℓ decays only as ∝ 𝑇𝑇 ⁻¹ , slower than ℓ ee , which should in principle allow the hydrodynamic description over a certain temperature range, until other phonon-mediated processes become important. So far, there has been little evidence for hydrodynamic electron transport. An exception is an early work on 2D electron gases in ballistic devices (ℓ ~ 𝑊𝑊) made from GaAlAs heterostructures (15). They exhibited nonmonotonic changes in differential resistance as a function of a large applied current 𝐼𝐼 that was used to increase the electron temperature (making ℓ ee short) while the lattice temperature remained low (allowing long ℓ).

The nonmonotonic behavior was attributed to the Gurzhi effect, a transition between Knudsen (ℓ ee ≫ ℓ) and viscous electron flows (7,15). Another possible hint for electron hydrodynamics comes from one of the explanations (16,17) for the Coulomb drag measured between two graphene sheets at the charge neutrality point (CNP).

Here we address electron hydrodynamics by using a special measurement geometry (Fig. 1) that amplifies effects of the shear viscosity 𝜈𝜈 and, at the same time, minimizes a contribution from ballistic effects that can occur not only in the Knudsen regime but also for viscous flow in graphene. As shown in Figs. 1A-B, a viscous flow can lead to vortices appearing in the spatial distribution of the steady-state current. Such ‘electron whirlpools’ have a spatial scale 𝐷𝐷 𝜈𝜈 = √𝜈𝜈𝜏𝜏, which depends on electron-electron scattering through 𝜈𝜈 and on the electron system’s quality through 𝜏𝜏 (18). To detect the whirlpools, electrical probes should be placed at a distance comparable to 𝐷𝐷 𝜈𝜈 . By using single- and bi- layer graphene (SLG and BLG, respectively) encapsulated between boron nitride crystals (19-21), we could reach 𝐷𝐷 𝜈𝜈 of 0.3-0.4 µm due to high viscosity of graphene’s Fermi liquid and its high carrier mobility 𝜇𝜇 even at high 𝑇𝑇. To the best of our knowledge, such large 𝐷𝐷 𝜈𝜈 are unique to graphene but still necessitate submicron resolution to probe the electron backflow. To this end, we fabricated multiterminal Hall bars with narrow (≈ 0.3 µm) and closely spaced (≈ 1 µm) voltage probes (Fig. 1C and fig. S1). For details of device fabrication, we refer to Supporting Material (18).

All our devices were first characterized in the standard geometry by applying 𝐼𝐼 along the main channel

and using side probes for voltage measurements. A typical behavior of longitudinal conductivity 𝜎𝜎 𝑥𝑥𝑥𝑥 at

a few characteristic 𝑇𝑇 of interest is shown in Fig. 1D. At liquid-helium 𝑇𝑇, the devices exhibited

𝜇𝜇 ~ 10-50 m ² V ^-1 s ^-1 over a wide range of carrier concentrations 𝑛𝑛 ~ 10 ¹² cm ^-2 , and 𝜇𝜇 remained above 5

m ² V ^-1 s ^-1 up to room 𝑇𝑇 (fig. S2). Such 𝜇𝜇 allow ballistic transport with ℓ > 1 µm at 𝑇𝑇 < 300 K. On the

other hand, at 𝑇𝑇 ≥ 150 K ℓ ee decreases down to 0.1-0.3 µm over the same range of 𝑛𝑛 (22, 23 and

figs. S3-S4). This allows the essential condition for electron hydrodynamics (ℓ _{𝑒𝑒𝑒𝑒} ≪ 𝑊𝑊, ℓ ) to be satisfied

within this temperature range. If one uses the conventional longitudinal geometry of electrical

measurements, it turns out that viscosity has little effect on 𝜎𝜎 𝑥𝑥𝑥𝑥 (figs. S5-S7) essentially because the

flow in this geometry is uniform whereas the total momentum of the moving Fermi liquid is conserved in

electron-electron collisions (18). The only evidence for hydrodynamics we could find in the longitudinal

geometry was the Gurzhi effect that appeared as a function of the electron temperature controlled by

applying large 𝐼𝐼, similar to the observations of ref. 15 (fig. S8).

Fig. 1. Viscous backflow in doped graphene. (A,B) Calculated steady-state distribution of current injected through a narrow slit for a classical conducting medium with zero 𝜈𝜈 (A) and a viscous Fermi liquid (B)

. (C) Optical micrograph of one of our SLG devices. The schematic explains the measurement geometry for vicinity resistance. (D,E) Longitudinal conductivity 𝜎𝜎 𝑥𝑥𝑥𝑥 and 𝑅𝑅 V as a function of 𝑛𝑛 induced by applying gate voltage. 𝐼𝐼 = 0.3 µA; 𝐿𝐿 = 1 µm. The dashed curves in (E) show the contribution expected from classical stray currents in this geometry (18).

To reveal hydrodynamics effects, we employed the geometry shown in Fig. 1C. In this case, 𝐼𝐼 is injected through a narrow constriction into the graphene bulk, and the voltage drop 𝑉𝑉 V is measured at the nearby side contacts located at the distance 𝐿𝐿 ~ 1 µm away from the injection point. This can be considered as nonlocal measurements, although stray currents are not exponentially small (dashed curves in Fig. 1E). To distinguish from the proper nonlocal geometry (24), we refer to the linear-response signal measured in our geometry as “vicinity resistance”, 𝑅𝑅 V = 𝑉𝑉 V /𝐼𝐼. The idea is that, in the case of a viscous flow, whirlpools emerge as shown in Fig. 1B, and their appearance can then be detected as sign reversal of 𝑉𝑉 V , which is positive for the conventional current flow (Fig. 1A) and negative for viscous backflow (Fig. 1B). Fig. 1E shows examples of 𝑅𝑅 V for the same SLG device as in Fig. 1D, and other SLG and BLG devices exhibited similar behavior (18). One can see that, away from the CNP, 𝑅𝑅 V is indeed negative over a wide range of intermediate 𝑇𝑇, despite a significant offset expected due to stray currents. Figure 2 details our observations further by showing maps 𝑅𝑅 V (𝑛𝑛, 𝑇𝑇) for SLG and BLG. The two Fermi liquids exhibited somewhat different behavior reflecting their different electronic spectra but 𝑅𝑅 V

was negative over a large range of 𝑛𝑛 and 𝑇𝑇 for both of them. Two more 𝑅𝑅 V maps are provided in fig.

S9. In total, seven multiterminal devices with 𝑊𝑊 from 1.5 to 4 µm were investigated showing the

vicinity behavior that was highly reproducible for both different contacts on a same device and different

devices, independently of their 𝑊𝑊, although we note that the backflow was more pronounced for

devices with highest 𝜇𝜇 and lowest charge inhomogeneity.

Fig. 2. Vicinity resistance maps. (A, B) 𝑅𝑅 V (𝑛𝑛, 𝑇𝑇) for SLG and BLG, respectively; the same color coding for the 𝑅𝑅 V scale. The black curves indicate zero 𝑅𝑅 V . For each 𝑛𝑛 away from the CNP, there is a wide range of 𝑇𝑇 over which 𝑅𝑅 V is negative. All measurements presented in this work for BLG were taken with zero displacement between the graphene layers (18).

The same anomalous vicinity response could also be observed if we followed the recipe of (15) and used the current 𝐼𝐼 to increase the electron temperature. In this case, 𝑉𝑉 V changed its sign as a function of 𝐼𝐼 from positive to negative to positive again, reproducing the behavior of 𝑅𝑅 V with increasing 𝑇𝑇 of the cryostat (fig. S10). Comparing figs. S8 and S10, it is clear that the vicinity geometry strongly favors the observation of hydrodynamics effects so that the measured vicinity voltage changed its sign whereas in the standard geometry the same viscosity led only to relatively small changes in 𝑑𝑑𝑉𝑉/𝑑𝑑𝐼𝐼. We also found that the magnitude of negative 𝑅𝑅 V decayed rapidly with 𝐿𝐿 (fig. S11), in agreement with the finite size of electron whirlpools (see below).

It is important to mention that negative resistances can in principle arise from other effects such single-electron ballistic transport (ℓ 𝑒𝑒𝑒𝑒 ≫ ℓ) or quantum interference (18,20,24). The latter contribution is easily ruled out because quantum corrections rapidly wash out at 𝑇𝑇 > 20 K and have a random sign that rapidly oscillates as a function of magnetic field. Also, our numerical simulations using the Landauer-Büttiker formalism and the realistic device geometry showed that no negative resistance could be expected for the vicinity configuration in zero magnetic field (19,21). Nonetheless, we carefully considered any ‘accidental spillover’ of single-electron ballistic effects into the vicinity geometry from the point of view of experiment. The dependences of the negative vicinity signal on 𝑇𝑇, 𝑛𝑛, 𝐼𝐼 and the device geometry allowed us to unambiguously rule out any such contribution as discussed in Supplementary Material. For example, the single-electron ballistic phenomena should become more pronounced for longer ℓ (that is, with decreasing 𝑇𝑇 or the electron temperature and with increasing 𝑛𝑛), in stark contrast to the nonmonotonic behavior of 𝑉𝑉 V .

Now we turn to theory and show that negative 𝑅𝑅 V arises naturally from whirlpools that appear in a viscous Fermi liquid near current-injecting contacts. As discussed in (18), electron transport for sufficiently short ℓ ee can be described by the hydrodynamic equations

∇ ⋅ 𝑱𝑱(𝒓𝒓) = 0 (1) and

𝜎𝜎 0

𝑒𝑒 ∇ϕ(𝒓𝒓) + 𝐷𝐷 _𝜈𝜈 ² ∇ ² 𝑱𝑱(𝒓𝒓) − 𝑱𝑱(𝒓𝒓) = 0 (2)

where 𝑱𝑱(𝒓𝒓) = 𝑛𝑛𝒗𝒗(𝒓𝒓) is the (linearized) particle current density, and ϕ(𝒓𝒓) is the electric potential in the 2D plane. If 𝐷𝐷 𝜈𝜈 → 0, Eq. (2) yields Ohm’s law, −𝑒𝑒𝑱𝑱(𝒓𝒓) = 𝜎𝜎 ₀ 𝑬𝑬(𝒓𝒓) with a Drude-like conductivity, 𝜎𝜎 ₀ ≡ 𝑛𝑛𝑒𝑒 ² 𝜏𝜏/𝑚𝑚 where −𝑒𝑒 and 𝑚𝑚 are the electron charge and the effective mass, respectively. The three terms in Eq. (2) describe: the electric force generated by the steady-state charge distribution in response to applied current 𝐼𝐼, the viscous force (1,2), and friction due to momentum non-conserving processes parametrized by the scattering time 𝜏𝜏(𝑛𝑛, 𝑇𝑇).

Fig. 3. Whirlpools in electron flow. (A-C) Calculated 𝑱𝑱(𝒓𝒓) and ϕ(𝒓𝒓) for the geometry such as that in Fig. 1C, with the green bars indicating voltage contacts. The color scale is for ϕ (red to blue corresponds to ±𝐼𝐼/𝜎𝜎 0 ). 𝐷𝐷 𝜈𝜈 = 2.3, 0.7 and 0 µm for panels A to C, respectively. Vortices are seen in the top right corners of A and B where the current flow is in the direction opposite to that in (C) that shows the case of zero viscosity. In each panel, the current streamlines also change color from white to black indicating that the current density |𝑱𝑱(𝒓𝒓)| is lower to the right of the injecting contact.

Equations (1,2) can be solved numerically as detailed in (18), and Fig. 3 shows examples of spatial distributions of ϕ(𝒓𝒓) and 𝑱𝑱(𝒓𝒓). One can clearly see that, for experimentally relevant values of 𝐷𝐷 𝜈𝜈 , a vortex appears in the vicinity of the current-injecting contact. This is accompanied by the sign reversal of ϕ(𝒓𝒓) at the vicinity contact on the right of the injector, which is positive in Fig. 3C (no viscosity) but becomes negative in Figs. 3A,B. Our calculations for this geometry reveal that 𝑅𝑅 V is negative for 𝐷𝐷 _𝜈𝜈 ≳ 0.4 µm (18). Because both 𝜏𝜏 and 𝜈𝜈 decrease with increasing 𝑇𝑇, 𝐷𝐷 _𝜈𝜈 also decreases, and stray currents start to dominate the vicinity response at high 𝑇𝑇. This explains why 𝑅𝑅 V in Figs. 1-2 becomes positive close to room 𝑇𝑇, even though our hydrodynamic description has no high-temperature cutoff.

Note that despite positive 𝑅𝑅 V the viscous contribution remains quite significant near room 𝑇𝑇 (Fig. 1D,

fig. S12). On the other hand, at low 𝑇𝑇 the electron system approaches the Knudsen regime and our

hydrodynamic description becomes inapplicable because ℓ 𝑒𝑒𝑒𝑒 ~ ℓ (18). In the latter regime, the

whirlpools should disappear and 𝑅𝑅 V become positive, in agreement with the experiment and our

numerical simulations based on the Landauer-Büttiker formalism.

The numerical results in Fig. 3 can be understood if we rewrite Eqs. (1,2) as

𝑱𝑱(𝒓𝒓) = ^𝜎𝜎 _𝑒𝑒 ⁰ ∇ϕ(𝑟𝑟) − 𝑛𝑛𝐷𝐷 _𝜈𝜈 ² ∇ × 𝝎𝝎(𝒓𝒓) , (3) where 𝝎𝝎(𝒓𝒓) ≡ 𝑛𝑛 ⁻¹ ∇ × 𝑱𝑱(𝒓𝒓) = 𝜔𝜔(𝒓𝒓)𝒛𝒛� is the vorticity (2). Taking the curl of Eq. (3), the vorticity satisfies the equation 𝜔𝜔(𝒓𝒓) = 𝐷𝐷 𝑣𝑣 2 ∇ ² 𝜔𝜔(𝒓𝒓) where 𝐷𝐷 _𝑣𝑣 plays the role of a diffusion constant. Current 𝐼𝐼 injects vorticity at the source contact, which then exponentially decays over the length scale 𝐷𝐷 𝑣𝑣 . For 𝐿𝐿 =1 µm, 𝜈𝜈 = 0.1 m ² s ^-1 and 𝜏𝜏 = 1.5 ps (18), we find 𝐷𝐷 𝑣𝑣 ≈ 0.4 µm, in qualitative agreement with the measurements in fig. S11.

Fig. 4. Viscosity of the Fermi liquids in graphene. (A,B) Solid curves: 𝜈𝜈 extracted from the experiment for SLG and BLG, respectively. Dashed: Calculations based on many-body diagrammatic perturbation theory (no fitting parameters). The grey-shaded areas indicate regions around the CNP where our hydrodynamic model is not applicable (18).

Finally, we combine the measurements of 𝑅𝑅 V and 𝜌𝜌 𝑥𝑥𝑥𝑥 with the solution of Eqs. (1,2) in Fig. 3 to extract the kinematic viscosity for SLG and BLG. Because the observed Gurzhi effect in 𝜌𝜌 𝑥𝑥𝑥𝑥 is small at low currents (fig. S6), we can use 𝜌𝜌 𝑥𝑥𝑥𝑥 = 1 𝜎𝜎 ⁄ = 𝑚𝑚/(𝑛𝑛𝑒𝑒 0 ² 𝜏𝜏) to determine 𝜏𝜏(𝑛𝑛, 𝑇𝑇) (18). Furthermore, for the experimentally relevant values of 𝐷𝐷 𝜈𝜈 , we find that 𝑅𝑅 V is a quadratic function of 𝐷𝐷 𝜈𝜈

𝑅𝑅 V = (𝑏𝑏 + 𝑎𝑎 𝐷𝐷 𝜈𝜈 2 )𝜎𝜎 0 −1 (4)

where 𝑎𝑎 and 𝑏𝑏 are numerical coefficients dependent only on the measurement geometry and

boundary conditions and 𝑏𝑏 describes the contribution from stray currents (fig. S13). For the specific

device in Fig. 3, we determine 𝑎𝑎 = −0.29 µm ^-2 and 𝑏𝑏 = 0.056, and this allows us to estimate 𝐷𝐷 𝜈𝜈 (𝑛𝑛, 𝑇𝑇)

from measurements of 𝑅𝑅 V . For the known 𝜏𝜏 and 𝐷𝐷 𝜈𝜈 , we find ν(𝑛𝑛, 𝑇𝑇) = 𝐷𝐷 𝜈𝜈 2 ⁄ . Applicability limits of 𝜏𝜏

this analysis are discussed in Supplementary Material, and the results are plotted in Fig. 4 for one of our

devices. It shows that, at carrier concentrations ~ 10 ¹² cm ^-2 the Fermi liquids in both SLG and BLG are

highly viscous with 𝜈𝜈 ≈ 0.1 m ² s ^-1 . For the sake of comparison, liquid honey has typical viscosities of

≈ 0.002-0.005 m ² s ^-1 .

Fig. 4 also plots results of fully-independent microscopic calculations of ν(𝑛𝑛, 𝑇𝑇), which were carried out by extending the many-body theory of ref. (25) to the case of 2D electron liquids hosted by SLG and BLG.

Within the range of applicability of our analysis in Fig. 4 (𝑛𝑛 ~ 10 ¹² cm ^-2 ), the agreement in absolute values of the electron viscosity is remarkable, especially taking into account that no fitting parameters were used in the calculations. Because the strong inequality ℓ ≫ ℓ 𝑒𝑒𝑒𝑒 required by the hydrodynamic theory cannot be reached even for graphene, it would be unreasonable to expect better agreement (18). In addition, our analysis does not apply near the CNP because the theory neglects contributions from thermally-excited carriers, spatial charge inhomogeneity and coupling between charge and energy flows, which can play a substantial role at low doping (16,18). Further work is needed to understand electron hydrodynamics in the intermediate regime ℓ ≳ ℓ 𝑒𝑒𝑒𝑒 and, for example, explain ballistic transport (ℓ > 𝑊𝑊) in graphene at high 𝑇𝑇 in terms of suitably modified hydrodynamic theory. Indeed, the naïve single-particle description that is routinely used for graphene’s ballistic phenomena even above 200 K (19,21) cannot be justified and needs to be explained in terms of electron-liquid jets. As for experiment, the highly viscous Fermi liquids in graphene and their accessibility offer a tantalizing prospect of using various scanning probes for visualization and further understanding of electron hydrodynamics.

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Acknowledgments: This work was supported by the European Research Council, the Royal Society, Lloyd’s Register Foundation, the Graphene Flagship and the Italian Ministry of Education, University and Research through the program Progetti Premiali 2012 (project ABNANOTECH). D.A.B. and I.V.G.

acknowledges Marie Curie program grant SPINOGRAPH.

Supplementary Material

#1 Device fabrication

Our devices were made from single- and bi-layer graphene encapsulated between relatively thick (~50 nm) crystals of hexagonal boron nitride (hBN). The crystals’ transfers were carried out using the dry-peel technique described previously (20,26). The heterostructures were assembled on top of an oxidized Si wafer (300 nm of SiO ² ) which served as a back gate, and then annealed at 300°C in Ar-H ² atmosphere for 3 hours. After this, a PMMA mask was fabricated on top of the hBN-graphene-hBN stack by electron-beam lithography. This mask was used to define contact areas to graphene, which was done by dry etching with fast selective removal of hBN (27). Metallic contacts (usually, 5 nm of Ta followed by 50 nm Nb) were then deposited onto exposed graphene edges that were a few nm wide. Such quasi-one-dimensional contacts to graphene (27) had notably lower contact resistance than those reported previously without the use of selective hBN etching (20). As the next step, another round of electron-beam lithography was used to prepare a thin metallic mask (≈ 40 nm Al) which defined a multiterminal Hall bar. Subsequent plasma etching translated the shape of the metallic mask into encapsulated graphene (see figs. S1A-B and Fig. 1C of the main text). The Al mask could also serve as a top gate, in which case Al was wet-etched near the Nb/Ta leads to remove the electrical contact to graphene. All our bilayer graphene (BLG) devices were prepared with such a top gate, which allowed us to control not only the carrier concentration but also the displacement field between the two layers.

Also, for single-layer graphene (SLG) we usually (but not always) made both top and bottom gates for the sake of fabrication procedures, even though the two gates fulfilled essentially the same function.

The studied Hall bars were 1.5 to 4 μm in width 𝑊𝑊 and up to 20 μm in length (larger 𝑊𝑊 were avoided as we previously found them to suffer from charge inhomogeneity induced by contamination bubbles and associated strain; ref. 28). The devices were carefully characterized and, in addition to Fig.

1D of the main text, an example of typical behavior of 𝜌𝜌 𝑥𝑥𝑥𝑥 (𝑛𝑛) is shown in fig. S1C. All the studied

devices, independently of their width or length, were found to exhibit negative vicinity resistance over

the described range of temperatures below room 𝑇𝑇 and over a wide range of 𝑛𝑛 ∼ 10 ¹² cm ^-2 . Fig. S1D

shows another example of this behavior, which is rather similar to that in Fig. 1E of the main text.

Fig. S1. Further examples of the studied graphene devices and their behavior. (A) Optical micrograph for an encapsulated SLG device. The bright white area is the top gate and the graphene Hall bar repeats its shape. Numerous metallic leads terminated with quasi-one-dimensional contacts to graphene are seen in a duller white color. Other colors on the photo appeared due to different etching depths of the hBN-graphene-hBN stack. (B) Electron micrograph of yet another SLG device. Resistivity (C) and vicinity resistance (D) for the device shown in (A). For resistivity measurements, we always used voltage probes separated by a distance larger than the main channel width. In (C), voltage probes were 8 µm away from each other. The vicinity probe used in (D) was 1 µm away from the current injecting lead. Positive and negative sign of 𝑛𝑛 correspond to gate-induced electrons and holes, respectively. The dashed curves in (D) show the expected ‘classical’ contribution 𝑏𝑏𝜌𝜌 𝑥𝑥𝑥𝑥 which arises due to stray currents. For this particular device, we find 𝑏𝑏 ≈ 0.1 using numerical simulations of the device geometry as described in the main text and the supplementary section on numerical simulation.

#2 Mobility and scattering times

Our longitudinal measurements allowed us to determine 𝜇𝜇(𝑛𝑛, 𝑇𝑇) and 𝜏𝜏(𝑛𝑛, 𝑇𝑇) using the standard

relation, 𝜎𝜎 𝑥𝑥𝑥𝑥 = 𝑛𝑛𝑒𝑒𝜇𝜇 = 𝑛𝑛𝑒𝑒 ² 𝜏𝜏/𝑚𝑚. Results are shown in fig. S2 for both SLG and BLG. The plotted behavior

is universal, that is, it changes little between different devices because, for the shown 𝑇𝑇 range of

interest, electron transport was limited by electron-phonon scattering. One can see that away from the

charge neutrality point (CNP), 𝜏𝜏 depends weakly on 𝑛𝑛 for both SLG and BLG. Typical times are of about

1-2 ps. As for 𝜇𝜇(𝑛𝑛, 𝑇𝑇), its behavior as a function of 𝑛𝑛 is notably different in the two graphene systems

because of different energy dependences of their effective masses. For BLG, which has a nearly parabolic

spectrum, we can for simplicity use the constant 𝑚𝑚 = 0.03𝑚𝑚 0 where 𝑚𝑚 0 is the free electron mass.

This yields that 𝜇𝜇 is simply proportional to 𝜏𝜏. For SLG, the effective (or cyclotron) mass is given by 𝑚𝑚 ∝ √𝑛𝑛, leading to 𝜇𝜇 varying approximately as 𝑛𝑛 ^−1/2 (fig. S2B).

Fig. S2. Phonon-limited transport in graphene. (A,B) Mean free times and mobilities in encapsulated SLG, respectively. (C,D) Same for encapsulated BLG. The plots are for the 𝑇𝑇 range in which hydrodynamics effects were found strongest.

#3 Microscopic calculations of the electron-electron mean free path

In this Section we briefly summarize the results of many-body diagrammatic perturbation theory calculations of the electron-electron scattering length ℓ ee . Results in this Section refer to SLG, in the region of parameter space relevant for our experiments.

We calculated ℓ ee = 𝑣𝑣 F 𝜏𝜏 ee from the imaginary part of the retarded quasiparticle self-energy 𝛴𝛴 𝜆𝜆 (𝑘𝑘, ω), evaluated at the Fermi surface. For an electron doped system we find ℏ 𝜏𝜏 ⁄ ee = −2 ℑ𝑚𝑚[𝛴𝛴 _𝜆𝜆=+1 (𝑘𝑘 F , 0)].

Here, 𝑣𝑣 F is the (bare) Fermi velocity (which is equal to the Dirac velocity 𝑣𝑣 D in SLG and ℏ𝑘𝑘 F /𝑚𝑚 in

BLG), 𝜆𝜆 = ±1 is a conduction/valence band index, and 𝜏𝜏 ee is the quasiparticle lifetime due to e-e

scattering (14). The quantity 𝛴𝛴 𝜆𝜆 (𝑘𝑘, ω) can be calculated by using the 𝐺𝐺 0 𝑊𝑊 approximation (23) with a

dynamically screened interaction 𝑊𝑊 𝒌𝒌,𝜔𝜔 evaluated at the level of the random phase approximation (RPA)

(14).

In the case of SLG, the imaginary part of the quasiparticle self-energy is given by the following expression (23)

ℑ𝑚𝑚[𝛴𝛴 _𝜆𝜆 (𝑘𝑘, 𝜔𝜔)] = − � � 𝑑𝑑 ² 𝒒𝒒

(2𝜋𝜋) ² ℑ𝑚𝑚 �𝑊𝑊 𝒒𝒒,𝜔𝜔−𝜉𝜉 _{𝜆𝜆′,𝒌𝒌+𝒒𝒒} � 𝐹𝐹 _{𝜆𝜆𝜆𝜆} ^′ [𝑛𝑛 _B �ℏ𝜔𝜔 − 𝜉𝜉 _𝜆𝜆 ^′ _{,𝒌𝒌+𝒒𝒒} � + 𝑛𝑛 _F �−𝜉𝜉 _𝜆𝜆 ^′ _{,𝒌𝒌+𝒒𝒒} �]

𝜆𝜆 ^′ =±1

where 𝐹𝐹 𝜆𝜆𝜆𝜆 ^′ = �1 + 𝜆𝜆𝜆𝜆 ^′ cos�𝜃𝜃 𝒌𝒌,𝒌𝒌+𝒒𝒒 ��/2 is the chirality factor, 𝜉𝜉 _{𝜆𝜆,𝒌𝒌} = 𝜆𝜆ℏ𝑣𝑣 _F 𝑘𝑘 − 𝜇𝜇 is the Dirac band energy measured with respect to the chemical potential 𝜇𝜇, 𝑊𝑊 𝒒𝒒,𝜔𝜔 = 𝑣𝑣 _𝑞𝑞 /𝜀𝜀(𝑞𝑞, 𝜔𝜔) ≡ 𝑣𝑣 _𝑞𝑞 /�1 − 𝑣𝑣 𝑞𝑞 𝜒𝜒 ₀ (𝑞𝑞, 𝜔𝜔)�

the RPA dynamically screened interaction, and 𝑛𝑛 B/F (𝑥𝑥) ≡ 1/[exp (𝛽𝛽𝑥𝑥) ∓ 1] are the Bose/Fermi statistical factors with 𝛽𝛽 = 1/(𝑘𝑘 B 𝑇𝑇). In the above expressions, 𝑣𝑣 _𝑞𝑞 is a suitably-chosen effective Coulomb interaction (see below), and 𝜒𝜒 0 (𝑞𝑞, 𝜔𝜔) is the polarization function of a non-interacting 2D massless Dirac fermion system at a finite temperature and carrier density (29). More details can be found, for example, in Refs. (22,23).

In our calculations we have also estimated the impact of the ‘environment’ such as i) nearby metal gates (by modeling them as perfect conductors), ii) the uniaxial anisotropy of dielectric hBN, and iii) thin-film effects. The bare Coulomb potential 2 𝜋𝜋𝑒𝑒 ² /𝑞𝑞 is strongly modified by these three factors. The effective Coulomb interaction 𝑣𝑣 𝑞𝑞 can be written in the form 2 𝜋𝜋𝑒𝑒 ² 𝒢𝒢(𝑞𝑞𝑑𝑑, 𝑞𝑞𝑑𝑑 ^′ )/𝑞𝑞 where the explicit functional dependence of the form factor 𝒢𝒢(𝑥𝑥, 𝑦𝑦) on its variables 𝑥𝑥 and 𝑦𝑦 is rather cumbersome and will be reported elsewhere. The form factor depends on the thickness 𝑑𝑑 and 𝑑𝑑 ^′ of the hBN slab below and above graphene, respectively. It also depends on the static values of the in-plane 𝜖𝜖 𝑥𝑥 (𝜔𝜔) and out-of-plane 𝜖𝜖 𝑧𝑧 (𝜔𝜔) components of the hBN permittivity tensor: see, for example, Ref. (30). Numerical results for ℓ ee in encapsulated SLG are shown in fig. S3.

Fig. S3. Numerical results for the e-e mean free path 𝓵𝓵 ee in our encapsulated SLG devices. Results are

shown as a function of carrier density 𝑛𝑛 and for three 𝑇𝑇. For this particular calculation, we used

𝑑𝑑 = 80 nm and 𝑑𝑑′ = 70 nm and took into account the top metal gate simulating the device shown in

fig. S1A. We have also checked that metal gates at such distances play little role (the presence of the

gate changed ℓ ee typically by less than 5% with respect to the ungated case), in agreement with the fact

that the SLG devices with and without top gates exhibited the 𝑅𝑅 V behavior indistinguishable within

variations between different contacts.

Besides determining the region of parameter space where the hydrodynamics theory can be applied, the frequency of electron-electron collisions also determines the numerical value of the electron liquid viscosity. The usual estimate for the value of the kinematic viscosity of a classical liquid is 𝜈𝜈 ∼ 𝑣𝑣ℓ coll (31), where 𝑣𝑣 is a characteristic velocity (e.g. the thermal velocity for classical liquids) of particles and ℓ coll

is the mean free path for inter-particle collisions.

Microscopic calculations for SLG yield (25):

𝜈𝜈 = ¹ ₄ 𝑣𝑣 F ℓ� ee (S1).

where ℓ� ee is a characteristic length associated with electron-electron scattering, which is of the same order of magnitude as ℓ ee in the explored range of parameters. Their ratio ℓ� ee /ℓ ee is shown in fig. S4.

Eq. (S1) is consistent with the above estimate for classical fluids. From Eq. (S1) we also find that the viscosity diffusion length 𝐷𝐷 𝜈𝜈 = √𝜈𝜈𝜏𝜏, which determines the size of electron whirlpools, is equal to 𝐷𝐷 _𝜈𝜈 = �ℓ� ee ℓ/2 and, therefore, depends on both electron-electron collisions and momentum-non-conserving collisions.

Fig. S4. Comparison between 𝓵𝓵 ee and 𝓵𝓵� ee . Numerical results for the ratio ℓ� ee /ℓ ee in the same range of densities and for the same temperatures as in fig. S3.

#4 On pseudo-relativistic and pressure terms in the Navier-Stokes equation

Because of the pseudo-relativistic nature of transport in SLG, the Navier-Stokes equation for the two-dimensional electron liquid in SLG contains a number of pseudo-relativistic terms (32). Such terms have not been considered in Eq. (2) of the main text. It is possible to demonstrate that, if one considers only linear deviations from a situation of uniform and static equilibrium (𝑛𝑛(𝒓𝒓, 𝑡𝑡) = 𝑛𝑛 and 𝒗𝒗(𝒓𝒓, 𝑡𝑡) = 𝟎𝟎), the only pseudo-relativistic correction that survives is the appearance of the effective (cyclotron) mass 𝑚𝑚 = ℏ𝑘𝑘 F ⁄ in the definition of the Drude-like conductivity 𝜎𝜎 𝑣𝑣 _{D 0} for the case of SLG.

In deriving Eq. (2) of the main text we have also neglected a term arising from the pressure 𝑃𝑃 of the

electron liquid, i.e. −∇𝑃𝑃(𝒓𝒓, 𝑡𝑡). Here we show that, for a gated structure like the one used in our

experiments, this term is simply proportional to the electric field and its only effect is to give a small

correction to the capacitance between the graphene sheet and the top and bottom gates. In a gated structure, the electric potential and carrier density can be related by the so-called local capacitance approximation (LCA) (32), i.e. ϕ(𝒓𝒓, 𝑡𝑡) = −𝑒𝑒𝑛𝑛(𝒓𝒓, 𝑡𝑡)/𝐶𝐶 where 𝐶𝐶 is the capacitance per unit area. For a double gated device, 𝐶𝐶 = 𝜖𝜖(𝑑𝑑 + 𝑑𝑑 ^′ ) (4 𝜋𝜋𝑑𝑑 𝑑𝑑′) ⁄ where 𝑑𝑑 and 𝑑𝑑′ are the distances between graphene and the bottom and top gates, respectively, and 𝜖𝜖 = �𝜖𝜖 𝑥𝑥 (0)𝜖𝜖 _𝑧𝑧 (0) ≈ 4.4 is the static dielectric constant of bulk hBN (we neglect the thin-film effects discussed in the previous Section). The quantities 𝜖𝜖 𝑥𝑥 (𝜔𝜔) and 𝜖𝜖 𝑧𝑧 (𝜔𝜔) have been introduced in the previous Section.

Using the LCA and the local density approximation ∇𝑃𝑃(𝒓𝒓, 𝑡𝑡) ≈ (𝜕𝜕𝑃𝑃 hom ⁄ )∇𝑛𝑛(𝒓𝒓, 𝑡𝑡), we can estimate the 𝜕𝜕𝑛𝑛 sum of the electric force and the force due to pressure as following

− � ^𝑒𝑒 _𝐶𝐶 ^{2 𝑛𝑛} + ^{𝜕𝜕𝑃𝑃} _{𝜕𝜕𝑛𝑛} ^hom � ∇𝑛𝑛(𝒓𝒓, 𝑡𝑡) (S2) where 𝑃𝑃 hom is the pressure of the homogeneous 2D electron liquid in SLG or BLG. Evaluating the two terms inside the round brackets at the equilibrium density 𝑛𝑛 and approximating 𝜕𝜕𝑃𝑃 hom ⁄ with its 𝜕𝜕𝑛𝑛 zero-temperature non-interacting value, i.e., 𝜕𝜕𝑃𝑃 hom ⁄ 𝜕𝜕𝑛𝑛 ≈ 𝜉𝜉𝐸𝐸 F where 𝜉𝜉 = 1 2 ⁄ (𝜉𝜉 = 1) for SLG (BLG), we can show that the ratio between the pressure term and the potential term is

𝜕𝜕𝑃𝑃 hom ⁄ 𝜕𝜕𝑛𝑛

𝑒𝑒 ² 𝑛𝑛 𝐶𝐶 ⁄ ≈ _{4𝑑𝑑𝑑𝑑} ^{𝑑𝑑+𝑑𝑑} _{′ 𝜉𝜉𝑘𝑘} ^′

TF (S3)

where 𝑘𝑘 TF is the Thomas-Fermi screening wave number. For encapsulated SLG, 𝑘𝑘 TF = 4 𝛼𝛼 ee 𝑘𝑘 F ≈ (2.9 nm) ⁻¹ , where 𝛼𝛼 ee = 𝑒𝑒 ² ⁄ (ℏ𝑣𝑣 D 𝜖𝜖) ≈ 0.5 is the so-called graphene fine structure constant (29).

Using a carrier density of 10 ¹² cm ⁻² , 𝑑𝑑 = 80 nm and 𝑑𝑑 ^′ = 80 nm, we find that the ratio in Eq. (S3) is much smaller than unity. The pressure term can be safely neglected. For an encapsulated BLG sheet 𝑘𝑘 TF = 2𝑒𝑒 ² 𝑚𝑚 (ℏ ⁄ ² 𝜖𝜖) ≈ (3.8 nm) ⁻¹ , irrespective of density (14). Therefore, the ratio (S3) is also negligible in this case.

#5 Smallness of the Reynolds number

The validity of the linearized Navier-Stokes equation (Eq. (2) of the main text) relies on the smallness of the Reynolds number (1), a dimensionless parameter that depends on the sample geometry and controls the smallness of the nonlinear term [𝒗𝒗(𝒓𝒓, 𝑡𝑡) ⋅ ∇]𝒗𝒗(𝒓𝒓, 𝑡𝑡) in the convective derivative in the full Navier-Stokes equation with respect to the viscous term. In our case

� [𝒗𝒗(𝒓𝒓,𝑡𝑡)⋅∇]𝒗𝒗(𝒓𝒓,𝑡𝑡)

𝜈𝜈∇ ² 𝒗𝒗(𝒓𝒓,𝑡𝑡) � ≈ ^{|𝒗𝒗|𝑊𝑊} _𝜈𝜈 = _{𝑒𝑒𝑛𝑛𝜈𝜈} ^𝐼𝐼 ≡ ℛ (S4).

For a typical probing current 𝐼𝐼 = 10 ⁻⁷ A, 𝑊𝑊 = 1 µm and 𝑛𝑛 = 10 ¹² cm ⁻² , we estimate |𝒗𝒗| ∼

𝐼𝐼 (𝑒𝑒𝑛𝑛𝑊𝑊) ⁄ ≈ 10 ⁴ cm/s. The corresponding value of the Reynolds number is ℛ ∼ 10 ⁻³ ≪ 1 if using

𝜈𝜈 ∼ 10 ³ cm ² /s found theoretically (25) and in the experiment (Fig. 4 of the main text). Our linearized

approximation is therefore fully justified.

#6 On the boundary conditions for solid-state hydrodynamic equations

The hydrodynamic equations need to be accompanied by appropriate boundary conditions (BCs). If viscosity is negligible, the current is proportional to the gradient of the potential. In this case it is sufficient to solve the Laplace equation for the potential to obtain both potential and current spatial patterns. The BCs that the potential must obey at the boundaries of the sample are of two types. In regions of the boundary where no electrical contacts are present, the normal component of the current (that is the normal derivative of the potential in the non-viscous regime) must be zero. In the regions of the boundaries where an electrical contact is present, the potential immediately inside the sample must be equal to the electric potential of the contact. Since the sample is current biased, we fix the total current flowing from each contact instead of fixing the value of the potential at each contact. It can be shown using standard theorems on the Laplace equation that these BCs (Neumann outside the contacts and Dirichlet at the contacts) uniquely determine the solution of the problem.

In the general case of a viscous flow, Eq. (2) of the main text requires additional BCs on the tangential component of the current. Generally, edges exert friction on the 2D electron liquid. The corresponding force (per unit length) is given by (1)

𝐹𝐹 _𝑡𝑡 = 𝜖𝜖 _{𝑖𝑖𝑖𝑖} 𝑛𝑛� _𝑖𝑖 𝜎𝜎 _{𝑖𝑖𝑘𝑘} ^′ 𝑛𝑛� _𝑘𝑘 (S5).

In Eq. (S5), 𝜎𝜎 _{𝑖𝑖𝑘𝑘} ^′ is the 2D viscous stress tensor, i.e. 𝜎𝜎 _{𝑖𝑖𝑘𝑘} ^′ = 𝜂𝜂�𝜕𝜕 𝑖𝑖 𝑣𝑣 _𝑘𝑘 + 𝜕𝜕 _𝑘𝑘 𝑣𝑣 _𝑖𝑖 − 𝛿𝛿 _{𝑖𝑖𝑖𝑖} 𝜕𝜕 _𝑙𝑙 𝑣𝑣 _𝑙𝑙 �. In writing the previous expression for 𝜎𝜎 _{𝑖𝑖𝑘𝑘} ^′ we have set to zero the diagonal contribution that is proportional to the so-called bulk viscosity and negligible (25).

The frictional force is in general a function of the tangential velocity 𝑣𝑣 𝑡𝑡 = 𝜖𝜖 _{𝑖𝑖𝑖𝑖} 𝑛𝑛� _𝑖𝑖 𝑣𝑣 _𝑖𝑖 . For small velocities the force is simply proportional to the velocity leading to the BC

𝜖𝜖 _{𝑖𝑖𝑖𝑖} 𝑛𝑛� _𝑖𝑖 𝑛𝑛� _𝑘𝑘 (𝜕𝜕 _𝑖𝑖 𝑣𝑣 _𝑘𝑘 + 𝜕𝜕 _𝑘𝑘 𝑣𝑣 _𝑖𝑖 ) = 𝜖𝜖 _{𝑖𝑖𝑖𝑖} 𝑛𝑛� _𝑖𝑖 𝑣𝑣 _𝑖𝑖 ⁄ 𝑙𝑙 b (S6) where 𝑙𝑙 b is a characteristic length scale associated with boundary scattering. If this length is very small, Eq. (S6) reduces to the standard “no-slip” boundary condition commonly used in the description of classical liquids (1).

Fig. S5. Influence of boundary conditions. Calculated current density 𝑱𝑱(𝒓𝒓) and electric potential

ϕ(𝒓𝒓) for the same geometry and the same 𝐷𝐷 𝜈𝜈 = 0.7 µm as in Fig. 3B of the main text. The difference

is no-slip boundary conditions (𝑙𝑙 b =0 ) used in this figure whereas 𝑙𝑙 b = ∞ in the main text.

We do not know precisely the value of 𝑙𝑙 b but we know that the combined effect of friction and viscosity arising at the boundaries can lead to an anomalous temperature dependence of the longitudinal resistivity 𝜌𝜌 𝑥𝑥𝑥𝑥 which is known as the Gurzhi effect (7). As discussed in one of the following Sections, our experimental data for 𝜌𝜌 𝑥𝑥𝑥𝑥 exhibit a monotonic behavior as a function of 𝑇𝑇, up to our highest temperature and for all carrier densities. This behavior suggests that the Gurzhi effect is small. For this reason, we can use to a good approximation the following BCs (12)

𝜖𝜖 𝑖𝑖𝑖𝑖 𝑛𝑛� 𝑖𝑖 𝑛𝑛� 𝑘𝑘 (𝜕𝜕 𝑖𝑖 𝑣𝑣 𝑘𝑘 + 𝜕𝜕 𝑘𝑘 𝑣𝑣 𝑖𝑖 ) = 0 (S7) which assumes that 𝑙𝑙 b is larger than the characteristics length scales of the problem, 𝐷𝐷 𝑣𝑣 (vorticity diffusion length) and 𝑊𝑊 (width of our multiterminal devices). Physically, Eq. (S7) corresponds to a vanishing tangential force acting on a moving liquid (1). At high current densities, however, the friction from the boundaries can be enhanced with respect to the simple linear model in Eq. (S6). In this case the Gurzhi effect can be observed in the differential resistance (see below).

In fig. S5 we show that different values of 𝑙𝑙 b have little impact on the formation of whirlpools near current injecting contacts. The reader is urged to compare fig. S5 (no-slip boundary conditions) with Fig.

3B in the main text, which was obtained using the free-surface BCs (Eq. S7).

#7 Applicability limits for hydrodynamic description of electron transport in doped graphene

The focus of our report is on the doped regime because the situation near the CNP is severely complicated by the presence of thermally excited quasiparticles, electron-hole puddles (33) and the large electron wavelength. In addition, thermoelectric effects (energy flow) are also expected to play a significant role near the CNP, although they appear only in the second order with respect to applied current 𝐼𝐼 in zero magnetic field (see, for example, ref. 34).

Under realistic experimental conditions, one important limit is set by charge inhomogeneity that impacts the viscosity analysis presented in Fig. 4 of the main text. Indeed, Eq. (4) assumes that 𝜎𝜎 0 = 1/𝜌𝜌 _{𝑥𝑥𝑥𝑥} is constant whereas the inhomogeneity locally modifies conductivity and stray currents. The electron-hole asymmetry seen in the experimental plots for 𝑅𝑅 V and the associated asymmetry in Fig. 4 of the main text are not expected in theory, and this provides a qualitative indication of the best accuracy one can expect for the extracted values of 𝜈𝜈.

Our hydrodynamic theory suggests no high-𝑇𝑇 cutoff, at least up to temperatures at which optical phonon scattering starts playing a role. In fact, the theory smoothly converges with the standard Drude theory as viscosity tends to zero upon increasing 𝑇𝑇. However, there is a clear high-𝑇𝑇 cutoff on 𝑅𝑅 V

being negative. It is simply dictated by the two competing terms in Eq. (4) of the main text, which are

due to stray currents and viscous flow. After subtracting the stray-current contribution from the

measured vicinity resistance (see below), we find that the hydrodynamic term smoothly extend to high T

over the entire temperature range without any sign of cutoff.

On the other hand, the essential condition of electron hydrodynamics (ℓ ee ≪ ℓ) certainly fails at temperatures below 50 K where the phase breaking length in graphene (which is smaller than ℓ ee ) is known to reach a micrometer scale (see, e.g., ref. 35) and electron transport can be described in terms of single-particle ballistics (billiard-ball model). Our hydrodynamic theory does not capture the crossover (ℓ ee ~ ℓ) into this single-particle regime, and it remains to be investigated how strong the above inequality condition should be to allow the hydrodynamic description.

#8 Absence of the Gurzhi effect in longitudinal resistivity

Resistivity of an electron liquid is determined by interplay between bulk scattering (charged impurities, lattice vibrations, crystal defects, etc.), collisions at sample boundaries and e-e scattering (7,15). Bulk scattering normally increases with 𝑇𝑇. On the other hand, a combined effect of boundary and electron-electron scattering results in a contribution to 𝜌𝜌 𝑥𝑥𝑥𝑥 which increases with 𝑇𝑇 if ℓ ee ≫ 𝑊𝑊 (Knudsen regime) but decreases if the electron system enters the viscous flow regime, ℓ ee ≪ 𝑊𝑊. The transition between the two limits may result in a non-monotonic temperature dependence of 𝜌𝜌 𝑥𝑥𝑥𝑥 . This phenomenon is referred to as the Gurzhi effect (7,15). In reality, this effect is severely obscured by various bulk scattering mechanisms and expected to be weak (15).

Fig. S6. Temperature dependence of longitudinal resistivity. Left and right panels are for SLG and BLG devices, respectively. The T dependences are monotonic, although one can notice that the curves slightly bulge around 100 K, which we attribute to a small hydrodynamics contribution related to the Gurzhi effect, as discussed in the next section.

In fig. S6, we show typical measurements of 𝜌𝜌 𝑥𝑥𝑥𝑥 as a function of 𝑇𝑇 for our SLG and BLG devices at different carrier concentrations. The behavior of 𝜌𝜌 𝑥𝑥𝑥𝑥 (𝑇𝑇) is monotonic (no Gurzhi effect) even in the region of parameter space where electron-electron scattering is strong enough to cause the observed sign change in the vicinity geometry. This can be attributed to relative insensitivity of electron flow to boundary scattering in this simplest geometry of measurements as discussed in the preceding section.

Neglecting more subtle effects observed in the differential resistance (see the next section), the absence

of the Gurzhi effect in 𝜌𝜌 𝑥𝑥𝑥𝑥 (𝑇𝑇) justifies our choice of (free-surface) BCs described by Eq. (S7), in which

the force exerted by the boundary on the electron fluid flow is neglected.

Using the BC of Eq. (S7), we have solved numerically the linearized steady-state hydrodynamic equations for the longitudinal geometry and the results are plotted in fig. S7 for SLG and BLG. This figure shows that 𝜌𝜌 𝑥𝑥𝑥𝑥 depends only on the phenomenological scattering time 𝜏𝜏 in the Navier-Stokes equation (Eq.

(2) in the main text) and exhibits little dependence on 𝐷𝐷 𝜈𝜈 and, hence, the electron viscosity 𝜈𝜈. This is the reason why we can use 𝜌𝜌 𝑥𝑥𝑥𝑥 (𝑛𝑛, 𝑇𝑇) to find 𝜏𝜏(𝑛𝑛, 𝑇𝑇) and, more generally, why the previous literature on electron transport in graphene, which completely neglected high electron viscosity, does not require revision if the measurements were carried out in the standard longitudinal geometry.

Fig. S7. Numerical solutions of the linearized hydrodynamic equations in the longitudinal geometry. In these plots we show the calculated longitudinal resistivity 𝜌𝜌 𝑥𝑥𝑥𝑥 as a function of 𝐷𝐷 𝜈𝜈 (in µm) for SLG (left) and BLG (right). In solving the hydrodynamic equations we have utilized the free-surface BCs of Eq. (S7).

From these numerical results, we infer that 𝜌𝜌 𝑥𝑥𝑥𝑥 is simply equal to the inverse of the Drude-like conductivity 𝜎𝜎 0 ≡ 𝑛𝑛𝑒𝑒 ² 𝜏𝜏/𝑚𝑚.

#9 Gurzhi effect with increasing the electron temperature

Despite the absence of notable deviations in 𝜌𝜌 𝑥𝑥𝑥𝑥 (𝑇𝑇) from a monotonic behavior, evidence for the Gurzhi effect could clearly be observed in the longitudinal differential resistance 𝑑𝑑𝑉𝑉/𝑑𝑑𝐼𝐼 measured as a function of a large applied current 𝐼𝐼. The current increased the temperature of the electron system well above the graphene lattice temperature and cryostat’s temperature, 𝑇𝑇 . Accordingly, these measurements enhanced electron-electron scattering whereas electron-phonon scattering remained relatively weak. Therefore, the 𝑑𝑑𝑉𝑉/𝑑𝑑𝐼𝐼 curves can qualitatively be viewed as changes in 𝜌𝜌 𝑥𝑥𝑥𝑥 induced by increasing the electron temperature. Examples of the observed 𝑑𝑑𝑉𝑉/𝑑𝑑𝐼𝐼 as a function of 𝐼𝐼 are shown in fig. S8.

At carrier concentrations |𝑛𝑛| > 1 × 10 ¹² cm ^-2 we observed rather featureless 𝑑𝑑𝑉𝑉/𝑑𝑑𝐼𝐼 curves up to our

highest 𝐼𝐼 ≈ 300 μA (fig. S8A). For smaller | 𝑛𝑛 |, the behavior of 𝑑𝑑𝑉𝑉/𝑑𝑑𝐼𝐼 became strongly

nonmonotonic, which can be attributed to the Gurzhi effect (7,15). We interpret the observed

nonlinearity as follows (15). At low 𝑇𝑇 and low 𝐼𝐼, electron-electron scattering is weak (ℓ ee ≳ 𝑊𝑊), and we

are in the Knudsen-like regime where the viscosity is determined by scattering at the channel edges. In

this regime, resistivity grows with increasing the electron temperature (increasing 𝐼𝐼), similar to the case

of classical dilute gases. At higher 𝐼𝐼 (> 50 µA), the further increase in the electron temperature pushes

starts being ruled by internal electron viscosity. The transition between the two regimes is known to lead to a drop in flow resistivity, as first observed by Knudsen for classical gases and, more recently (15), reported as the Gurzhi effect for electrons, also using the 𝑑𝑑𝑉𝑉/𝑑𝑑𝐼𝐼 measurements. The 𝑇𝑇 dependence in fig. S8B shows that 𝐼𝐼~100 μA heats up the electron system to ~200 K, which leads to the transition into the Navier-Stokes regime. This is in good agreement with the 𝑇𝑇 range where our hydrodynamic effects were found strongest. Also, note that the Gurzhi effect appeared within the same range of carrier concentrations in which we observed largest negative 𝑅𝑅 V (compare fig. S8 with Fig. 2A of the main text and fig. S9).

Fig. S8. Longitudinal differential resistance. (A) Examples of 𝑑𝑑𝑉𝑉/𝑑𝑑𝐼𝐼 as a function of applied current for a SLG device. To measure 𝑑𝑑𝑉𝑉/𝑑𝑑𝐼𝐼, we applied an oscillating current 𝐼𝐼 + 𝐼𝐼 ac cos (𝜔𝜔𝑡𝑡) along the main channel where 𝐼𝐼 ac is the low-frequency current, much smaller than 𝐼𝐼. The ac voltage drop that appeared at side contacts yielded 𝑑𝑑𝑉𝑉/𝑑𝑑𝐼𝐼. The main channel was 2.5 µm wide, and voltage probes were separated by 8 µm. 𝑇𝑇 = 5 K; 𝐼𝐼 ac = 50 nA. (B) Temperature dependence of the differential resistance in (A) for hole doping with 𝑛𝑛 = −0.4 × 10 ¹² cm ^-2 . The curves in (B) are offset for clarity by 300 Ohms each.

#10 Reproducibility of negative vicinity response

To illustrate that the observed whirlpool effects were reproducible for different devices and using

different contacts, fig. S9 shows two more examples of 𝑅𝑅 V maps. They are for SLG devices with low-𝑇𝑇

𝜇𝜇 of ≈ 50 m ² V ^-1 s ^-1 and the distance 𝐿𝐿 to the nearest vicinity probe of ≈ 1 µm. These maps are

rather similar to those shown in Fig. 2 of the main text. Again, we observed large negative vicinity

resistance away from the CNP and over a large range of 𝑇𝑇 and 𝑛𝑛.

Fig. S9. Further examples of negative vicinity resistance. (A) and (B) are 𝑅𝑅 V (𝑛𝑛, 𝑇𝑇) maps for two different high-quality SLG devices and the distance between the injection and vicinity contacts of 1 µm.

#11 Changes from normal flow to backflow induced by electron heating

Negative vicinity voltage 𝑉𝑉 V is attributed to electron whirlpools and expected only in the viscous-flow regime. This requires ℓ ee to be smaller than the characteristic scale in the problem, that is, the distance 𝐿𝐿 between the injector and voltage contacts. In addition, to be detectable in transport experiments whirlpools should be sufficiently large in size (large 𝐷𝐷 𝑣𝑣 ) to reach from the injection region to voltage probes. Because ℓ ee depends on the electron temperature, the nature of electron flow can be controlled not only by changing the lattice temperature as in the experiments described in the main text but also by heating up the electron system using large dc currents 𝐼𝐼 as discussed in the above section on the Gurzhi effect. We have carried out such measurements of 𝑉𝑉 V as a function of the electron temperature, and examples of the observed negative vicinity response are shown fig. S10A. It plots typical behavior of 𝑉𝑉 V as a function of 𝐼𝐼 for three characteristic temperatures of the cryostat, T. For the case of low 𝑇𝑇, the 𝐼𝐼-𝑉𝑉 curve exhibits a positive slope at small 𝐼𝐼 which corresponds to the same linear-response 𝑅𝑅 V = 𝑉𝑉 V /𝐼𝐼 as in the maps of Fig. 2 of the main text and fig. S9. This is the Knudsen flow regime. At higher currents (𝐼𝐼 > 50 μA), the voltage response becomes nonlinear reaching first a maximum and then changing the sign to negative. This is because the current heats up the electron system and drives it into the Navier-Stokes regime such that whirlpools appear near the injection point.

At even higher currents, 𝑉𝑉 V changes its sign again, from negative to conventional positive, indicating

that the electron temperature becomes high enough (> 300 K) and the system approaches the high-𝑇𝑇

regime of small 𝐷𝐷 𝑣𝑣 . If we increased the cryostat temperature to 100 K (fig. S8A), the electron system

entered the viscous-flow regime even at vanishingly small probing currents, and the 𝐼𝐼-𝑉𝑉 curves – linear

over a large range of 𝐼𝐼 – yield negative 𝑅𝑅 V , in agreement with the results presented in the main text. At

sufficiently high currents, the system again exhibits positive 𝑉𝑉 V , which corresponds to dominating stray

currents. Further increase in 𝑇𝑇 in fig. S10A, changes the character of 𝐼𝐼-𝑉𝑉 characteristics once again

because the system is already close to the transport regime of small 𝐷𝐷 𝑣𝑣 , even without being heated by

current. Note that these changes are closely connected with the Gurzhi effect reported in fig. S8.

However, because the vicinity geometry is much more sensitive to a viscous flow contribution, voltage rather than its derivative changes the sign as a function of 𝐼𝐼 in fig. S10.

Fig. S10. Vicinity voltage as a function of applied current. (A) 𝐼𝐼-𝑉𝑉 characteristics at three characteristic 𝑇𝑇 for a BLG device at hole doping 𝑛𝑛 = −2.5 × 10 ¹² cm ^-2 . (B) Map of the normalized nonlinear vicinity resistance 𝑉𝑉 V /𝐼𝐼 measured at 𝑇𝑇 = 5 K. It is important to note that large 𝐼𝐼 can lead to temperature gradients between different contacts and, as a result, spurious thermoelectric signals may appear in such measurements. Because the thermopower contribution depends only on the absolute value of 𝐼𝐼 and not on its sign, the contribution can easily be eliminated by symmetrizing 𝑉𝑉 V with respect to the direction of dc current. This procedure was applied for the shown plots and, accordingly, they are presented as a function of |𝐼𝐼|. The brown rectangle in (B) is the region around the CNP with no collected data to avoid overheating and switching between different mesoscopic states.

For further comparison between effects of electron heating and cryostat’s 𝑇𝑇, fig. S10B shows a map of 𝑉𝑉 V /𝐼𝐼, the nonlinear vicinity response normalized by the applied current. This map closely resembles the 𝑅𝑅 _V (𝑇𝑇, 𝑛𝑛) maps in Fig. 2 of the main text and fig. S9 and also shows a clear transition from normal electron flow at low T to backflow at intermediate electron temperatures. Note that in fig. S10B we had to limit our measurements to relatively small 𝐼𝐼 < 200 µA so that the transport regime dominated by stray currents (approached above 400 µA in fig. S10A) could not be reached. This is because such high currents occasionally switched our devices between different mesoscopic states whereas the 𝑉𝑉 V maps required a few days of continuous measurements. For the same reason, we avoided measurements of 𝑉𝑉 V (𝐼𝐼) around the charge neutrality point in fig. S10B where high resistivity of graphene resulted in significant Joule heating even for relatively small currents.

The observed strong enhancement of the negative vicinity signal with increasing the electron

temperature is in good agreement with the expected behavior of local whirlpools inside graphene’s

electron liquid and, also, rules out a contribution from single-particle ballistic effects. Indeed, we found

experimentally that the latter phenomena such as negative transfer resistance and magnetic focusing

(19,21) are rapidly and monotonically suppressed with increasing 𝐼𝐼.

#12 Dependence of electron backflow on distance to the injection contact

We have investigated how negative vicinity resistance decays with increasing the distance 𝐿𝐿 between the injection and voltage contacts (see the sketch in fig. S11). This figure shows examples of the temperature dependence of 𝑅𝑅 V in the linear 𝐼𝐼-𝑉𝑉 regime for different 𝐿𝐿 = 1, 1.3 and 2.3 μm, which were measured for the same BLG device at a fixed carrier concentration of 1.5 × 10 ¹² cm ^-2 . All the plotted curves exhibit negative 𝑅𝑅 V but the temperature range in which the backflow occurs rapidly narrows with increasing 𝐿𝐿, and we could not detect any backflow for 𝐿𝐿 > 2.5 μm in any of our devices.

The magnitude of the negative signal is found to decay rapidly (practically exponentially) with 𝐿𝐿 (top inset of fig. S11), yielding a characteristic scale of ≈ 0.5 μm. This provides a qualitative estimate for the size of electron whirlpools, in agreement with our theoretical estimates for 𝐷𝐷 𝑣𝑣 . Indeed, for the particular device in fig. S11, we can estimate 𝐷𝐷 𝑣𝑣 ≈ 0.4 μm using our independent measurements of 𝜈𝜈 ≈ 0.1 m ² s ^-1 and 𝜏𝜏 ≈ 1.5 ps (see the main text and above).

Fig. S11. Vicinity resistance measured at different distances from the injection contact. All the contacts were ≈ 0.3 µm in width. Top inset: Maximum value of negative 𝑅𝑅 V as a function of 𝐿𝐿. The dashed curve is the best fit with 𝐷𝐷 𝑣𝑣 ≈ 0.5 μm. The probing current was 0.3 µA.

#13 Stray-current contribution to the vicinity resistance

In the vicinity geometry, stray currents near the voltage probe are not negligible. Their contribution to

the measured vicinity resistance is given by the first term 𝑏𝑏𝜎𝜎 0 −1 in Eq. (4) of the main text where b is

the geometrical factor dependent on L, W and width of the contact regions (36). Fig. 1E of the main text

and fig. S1D clearly show that the classical contribution was rather significant and competed with the

viscous term in 𝑅𝑅 𝑉𝑉 over a range of T and 𝑛𝑛. In this report, we have deliberately focused on the sign

change in 𝑅𝑅 𝑉𝑉 because the negative resistance is an exceptional qualitative effect, which in our case

cannot be explained without taking into account a finite viscosity of the electron liquid. However, to

elucidate the hydrodynamic behavior in more detail, we can go a step further and analyze the

anomalous part of 𝑅𝑅 𝑉𝑉 , which comes on top of the contribution from stray currents. To this end, we

exclusively due to a finite viscosity. Fig. S12 shows a typical example of ∆𝑅𝑅 V observed in our devices. It is clear that at T > 50 K the negative ∆𝑅𝑅 V extends over the entire range of carrier concentrations away from the CNP (fig. S12A). Figure S12B suggests that electron whirlpools persist well above room 𝑇𝑇.

It is important to note that the above subtraction analysis is based on the assumption of spatially uniform 𝜎𝜎 0 whereas the experimental devices exhibit a certain level of charge inhomogeneity, especially close to the CNP. Qualitatively, one can gauge the influence of charge inhomogeneity from the pronounced electron-hole asymmetry in the 𝑅𝑅 V maps, which in theory should be symmetric. The asymmetry was found to be contact dependent and arises due to non-uniform charge distribution near the vicinity contacts. Furthermore, the subtraction analysis is not applicable in the low-𝑇𝑇 regime because it ignores single-particle ballistic effects that modify stray currents on a distance of the order of the mean free path. Notwithstanding these limitations, the subtraction procedure in fig. S12 provides a qualitatively accurate picture, especially at high 𝑇𝑇 where single-particle phenomena can be neglected and for 𝑛𝑛 ≳ 1 × 10 ¹² cm ^-2 where the electron system become more uniform.

Fig. S12. Hydrodynamic part of vicinity resistance after subtracting a calculated contribution from stray currents. (A) ∆𝑅𝑅 𝑉𝑉 (𝑛𝑛) at two characteristic temperatures and (B) ∆𝑅𝑅 _𝑉𝑉 (𝑇𝑇) for a typical carrier density away from the neutrality point. (C) Map ∆𝑅𝑅 𝑉𝑉 (𝑛𝑛, 𝑇𝑇) covering the entire range of measured temperatures and concentrations. Data are for the same device as in Fig. 2B of the main text. The red traces outline the region of negative 𝑅𝑅 𝑉𝑉 in Fig. 2B. The brown rectangle indicates the region with ∆𝑅𝑅 𝑉𝑉

> +10 Ohm around the CNP where our hydrodynamic analysis is not expected to be applicable.